Abstract:
I will review recent results concerning the Ginzburg - Landau equations. These equations were first developed to understand macroscopic behaviour of superconductors; later, together with their non-Abelian generalizations - the Yang-Mills-Higgs equations, they became a key part of the standard model in elementary particle physics. They also have found important applications in geometry and topology.

The Ginzburg - Landau equations have remarkable solutions, localized topological solitons, called the magnetic vortices in the superconductivity and the Nielsen-Olesen or Nambu strings in the particle physics, as well as extended ones, magnetic vortex lattices.

I will review the existence and stability theory of the vortex lattice solutions and how they relate to the modified theta functions appearing in number theory and algebraic geometry. Certain automorphic functions play a key role in the theory described in the talk.